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This article is meant to give an exhaustive list of explicitly constructed nontopological functorial field theories in dimension 2 and higher. All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.
Probably one should count all rational 2d CFTs as examples, via the FRS construction.
Re #2: Can we pinpoint a specific theorem in their series of papers that actually constructs a (nontopological) functorial field theory?
It’s most explicit in the last article in the series:
In section 3.4 there they give a definition of (rational) functorial conformal field theory, then they proceed to show that their construction produces all these.
They formulate the 2d functorial field theory slightly differently than usual because their primary perspective is the functorial 3d Chern-Simons theory defined by the given modular tensor category, from which the 2d CFT is obtained in a “holographic fashion” as a twisted/boundary field theory of the 3d functorial field theory.
This way the eponymous “functor” underlying their functorial field theory is the “modular functor” which assigns vector spaces not to 1d boundaries but to cobordisms (these being the spaces of conformal blocks = the spaces of states of the ambient 3d CS theory) and then the 2d CFT proper appears to them as a natural transformation into that functor which picks in each such space an actual correlator — the latter being secretly the linear map that the 2d functorial field theory assigns to the given cobordism.
In this perspective, the functoriality of the 2d CFT is the “sewing constraints” satisfied by that natural transformation into the modular functor.
They are more explicit about this perspective on functorial CFT in Section 2 of the review:
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